### The impossibility of asymmetric causation

Apart from the innate symmetricity of structural equation systems, the very definition for the conditional probability as well as Bayes's law of inverse probability unambiguously suggest the reversibility of cause-effect relationships. …

“*A* is a cause of *B* only when *B* can cause *A*.'' Amen.

(Quoted from Mr. Asha review of *Causality*, amazon.com, August 29, 2000)

It is instructive to compare the apparent symmetries of Bayes rule and structural equations with the asymmetry inherent in out conception of causation. Bayes rule asserts that if

Ais relevant toB, thenBmust be relevant toA, that is, if learningAchanges the probability ofBthen learningBmust change the probability ofA. This symmetry constitutes the first of the five properties of graphoids (Causality, page 11). However, this symmetry refers to informational relevance, not to causal relevance (as axiomatized inCausality, Section 7.3.3). Bayes's rule and information relevance deal with changes in one's beliefs about a static world; causality deals with changes in the world itself. My belief about the rain may change in light of reading the barometer, but there is nothing I can do to the barometer that will change the rain in the physical world.The apparent symmetry of structural equations is equally deceiving. On page 160 I explain how structural equations serve a dual purpose, observational and interventional, and I argue that it is the inteventional component which distingushes structural equations from algebraic equations. Referring to the equation

y=bx+e, page 160 reads: "Note that the operational reading just given makes no claim

about how

X(or any other variable) will behave when we control

Y. This asymmetry makes the equality signs in structural equations different from algebraic equality signs;

the former act symmetrically in relating observations on

XandY (e.g., observing

Y=0 implies bx= –e), but they act asymmetrically when it comes to interventions

(e.g., setting

Yto zero tells us nothing about the relation between

xand e). The arrows in path diagrams make this dual role explicit, and this may account for the

insight and inferential power gained through the use of diagrams.

Although the literature on structural equation models does not explicitly acknowledge this basic interpretation of structural equations — a puzzling phenomenon that I explain in Section 5.1 — it is implicit in the conclusions that scientists draw from SEM studies.

Best wishes,

========Judea Pearl

Comment by judea — February 21, 2007 @ 11:49 pm